The two‐dimensional, free‐surface flow of a Newtonian fluid exiting from a planar die is computed by finite element analysis using quasiorthogonal mesh generation and local mesh refinement with irregular, embedded elements to obtain extreme resolution of the velocity and pressure fields near the die edge, where the fluid sheet attaches to the solid boundary. Calculations for the limit of large surface tension, the stick‐slip problem, reproduce the singular behavior near the die edge expected from asymptotic analysis using a self‐similar form for the velocity field. Results for finite capillary number (Ca) predict that the meniscus separates from the die at a finite contact angle and suggest that the capillary force enters the dominant normal stress balance at the die edge through an infinite curvature, as previously suggested by Schultz and Gervasio. The size of this region with large positive curvature increases with increasing Ca, and the strength of the singularity is in good agreement with theoretical predictions for a straight meniscus attached to the die at the appropriate contact angle predicted by the simulations. The contact angle appears to be determined from matching of the inner solution structure valid near the singularity with the bulk flow, in agreement with arguments made by Ramalingam; increasing the Reynolds number decreases the contact angle, corroborating this effect. Introducing fluid slip along the surface of the die changes the structure of the singularity in the pressure and stresses, but does not alleviate the singular behavior. In fact, the calculations with slip coefficients small enough not to change the bulk solution are more difficult than calculations with the no‐slip boundary condition. ©1995 American Institute of Physics.